Question

If the remainder when the polynomial f(x) is divided by (x-1) & (x+1) are 6 and 8 , find the remainder when f(x) is divided by (x2-1)

Answer 1

Answer:

The remainder will be (-x +7)

Step-by-step explanation:

when f(x) is divided by x-1 the remainder is 6, let the quotient be a

=> f(x) = a(x-1) + 6

=> f(1) = a(1-1) + 6 = 6

when f(x) is divided by x+1 the remainder is 8, let the quotient be b

=> f(x) = b(x+1) + 8

=>f(-1) = b(-1 + 1) + 8 = 8

when f(x) will be divided by x²-1 the remainder should be a function of x with degree less than 2

let the remainder be gx+h and the quotient be c

hence

f(x) = c(x²-1) + gx +h

=> f(1) = c(1-1) + g(1) + h = g+h = 6..............eqn 1

also,

f(-1) = c(1-1) + g(-1) + h = -g+h = 8.............eqn 2

solving both the eqn we get

g = -1

h =7

hence the remainder will be (-x +7)